you'd be better to stay on 29 to sioux falls then shoot west.
the route going north from north platte takes you through a wildlife refuge south of valentine. its a long boring drive through the sand hills and is 2 lane. its not actually all bad.
if you do take it make sure you fill up in valentine before going any further north. once you hit the border your on the rosebud indian reservation, theres 1 gas station in rosebud and thats it until the interstate and murdo.
make sure to stop in murdo and the car museum :) on your way to rapid.
most tourist attractions in the hills arent yet open, but some are year round.

do you plan on stopping at mt rushmore and crazy horse? if so stop at rushmore then backtrack a couple miles to hwy 16a (iron mt road) excellent drive and takes you to custer state park if you want to see buffalo and such. drive through to custer then head north a few miles to see crazy horse.
you will see so much more taking 16a including the pigtail bridges and one lane tunnels carved through the granite, not to mention a spectacular view of mt rushmore from the top. also a must drive if you want would be take 16a to 87 needles hwy. very excellent drive and scenery.
I can do you a map if youd like.